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# Logarithmic Functions

## 3.3 Differentiation Rules for Exponentials and Logarithms

#### Differentiation Rules for Exponentials and Logarithms

• The other rules regarding differentiation such as addition, subtraction, multiplication, division of functions, chain rule etc. applies too.
• All of these rules can be proven via first principles, but it is not required.
• The basic rules in this section are:

i) If f(x)=a^{x}, \text { then } f^{\prime}(x)=a^{x} \ln a

ii) If f(x)=\log _{a} x, \text { then } f^{\prime}(x)=\frac{1}{x}(\ln a)^{-1}

where n \neq 0, n \in R (or n \in R \backslash\{0\})

• Applying the rules to natural exponential and logarithm functions (i.e. a=e), yields:
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## 3.3 Logarithm Laws [Free]

#### The Logarithm Function’s Algebraic Properties

• There are four properties in logarithms to consider:

i) \log_{a}{mn}=\log_{a}{m}=\log_{a}{n}

ii) log_{a}{(\frac{m}{n})}=\log_{a}{m}-\log_{a}{n}

iii) log_{a}{(m^{p})}=p\cdot{\log_{a}{m}}

In particular when p=-1, we have log_{a}{(\frac{1}{m})}=-\log_{a}{m}

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## 5.3 Special Exponential and Logarithmic Functions

#### Special Exponential Functions

• The exponential function y=e^x is a special exponential function, and is vital in the field of mathematics.
• In particular, e=\lim _{n \rightarrow \infty}\left(1+\frac{1}{n}\right)^{n} is a constant known as the Euler’s number. It is approximately 2.718281459045…, and it is an irrational number like \pi.
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## 3.7 Transformations of Logarithmic Functions

#### Plotting Logarithmic Functions

Example

Say y=log_{e}{x}. Ideally, you should remember the general shape of the graph, then labelling any important points (i.e. Step 2).

Generally, the steps to plot a graph are:

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## 1.4 Logarithmic Functions

#### The Logarithmic Function

• The logarithmic function with base is defined as follows:

a^{x} = b is equivalent to log_{a}{b}=x, where a \in R^{+}\setminus \left \{ 1 \right \} .

Note:a \in R^{+}\setminus \left \{ 1 \right \}‘ means that could be any positive number, excluding 1.

• The expression log_{a}{y}=x is read as: ‘the logarithm of y to the base a is equal to x’.

Example

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## Derivatives

This tutorial covers material encountered in chapter 9 of the VCE Mathematical Methods Textbook, namely: The derivative of functions seen previously in tutorial worksheets 1-5… Read More »Derivatives

## Exponential and Logarithm Functions

This tutorial covers material encountered in chapter 5 of the VCE Mathematical Methods Textbook, namely: Exponential functions Index laws Log functions Log laws and change… Read More »Exponential and Logarithm Functions